Math 6420 • PDE

\begin{enumerate}
\item[(a)] Describe the Gram-Schmidt orthogonalization procedure for a set of \(n\)-vectors
\(\{u_1,u_2,\ldots,u_k\}\).
\item[(b)] Suppose \(k<n\) and that the vectors \(\{u_1,u_2,\ldots,u_k\}\) are linearly
independent. Show that the Gram-Schmidt procedure is equivalent to factoring the matrix
\(U\) with column vectors \(\{u_1,u_2,\ldots,u_k\}\) as \(U=QR\) where \(R\) is triangular. What is
the structure of \(Q\)?
\item[(c)] Use this factorization of \(U\) to find the least-squares solution of \(Ux=b\).
\end{enumerate}